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<title>Atlas software user guide -- Tori</title>
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<h2>Real algebraic tori</h2>
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<p>
<i>Last updated: April 10, 2005</i>
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<p>
The following data are equivalent:
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<ol>
<li>a complex algebraic torus T with involution &theta;</li>
<li>a real algebraic torus (see <a href="realforms.html">here</a>)</li>
<li>a lattice X with involution &theta;</li>
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<p>
It will be convenient to use 3. As explained in Fokko du Cloux's 2004
<a href="http://atlas.math.umd.edu/papers">notes</a>, it is always possible
to find a basis
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<p align="center">
e<sub>1</sub>, ... ,e<sub>p</sub>,&nbsp;
e<sub>p+1</sub>, ... ,e<sub>p+q</sub>,&nbsp;
e<sub>p+q+1</sub>, ... , e<sub>p+q+2r</sub>
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<p>
of X such that the vectors e<sub>1</sub>, ... ,e<sub>p</sub> are eigenvectors 
of &theta; for the eigenvalue +1, e<sub>p+1</sub>, ... ,e<sub>p+q</sub> are 
eigenvectors of theta for the eigenvalue -1, and the vectors e<sub>j</sub> and 
e<sub>j+r</sub> are interchanged by &theta; for p+q < j &le; p+q+r. 
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<p>
Here the numbers p, q and r are entirely determined:
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<ul>
<li>
p+r is the number of +1 eigenvalues of &theta; (say over the rational numbers);
</li>
<li>q+r is the number of -1 eigenvalues of &theta;</li>
<li>and r is computed as follows: &theta; induces a unipotent automorphism
of the <b>Z</b>/2-vector space X/2X; then r is the dimension of the image of
&theta;-Id in this vector space.
</li>
</ul>

It follows that the group of real points of T is isomorphic to the product
of p copies of the compact torus <b>U</b>(1), q copies of the multiplicative
group of the reals, and r copies of the multiplicative group of the complex
numbers. In particular, the component group of T(<b>R</b>) is isomorphic to
the product of q copies of <b>Z</b>/2.

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<p>
<a href="userintro.html">Back</a> to the introduction.<br>
<a href="http://atlas.math.umd.edu">Back</a> to the Atlas homepage.
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